zArc
plain-language theorem explainer
Specializes the Wick arc on the timelike squared edge to unit lattice spacing and unit scale factor: z(t) = exp(i π (1-t)). Lorentzian endpoint z(0) = -1, Euclidean endpoint z(1) = +1, interior in the open upper half-plane. Cited throughout the C11 hinge-data continuation (branch regularity, continuity, endpoint IDs). One-line wrapper of the general arc at a = α = 1.
Claim. For real parameter $t$, define the physical-point Wick arc $z_{\mathrm{arc}}(t) := \exp\bigl(i\pi(1-t)\bigr)\in\mathbb{C}$. Equivalently, it is the canonical upper-half-plane arc on the timelike squared edge at unit edge length and unit scale: $z_{\mathrm{arc}}(t) = \mathrm{arc}_Z(1,1,t)$.
background
Module C11 formalizes complex-first 4D Wick continuation of Regge hinge data (Cayley–Menger areas-squared and cofactor dihedral cosines) for causal 4-simplex classes. The continuation path on the timelike squared edge is the canonical upper-half-plane arc
$$z(t)=\alpha a^2\exp\bigl(i\pi(1-t)\bigr),\quad t\in[0,1],$$
with $z(0)=-\alpha a^2$ Lorentzian and $z(1)=+\alpha a^2$ Euclidean, matching lorentzianSqEdges / euclideanSqEdges sign convention. Interior $t\in(0,1)$ has strictly positive imaginary part.
The general constructor is arcZ a alpha t. This definition freezes the physical point used by the hour-0 numeric gate and by physicalCausalPent: unit lattice spacing $a=1$ and scale $\alpha=1$, so $|z|=1$ and $z(t)=e^{i\pi(1-t)}$. Spacelike edges stay real $a^2$; only timelike edges ride the arc. Scope is hinge-data continuation only; full action-level continuation remains the open C12 / ledger gap.
proof idea
Pure abbreviation: zArc t is defined as arcZ 1 1 t. No tactics. Unfolding arcZ immediately yields
$$((1\cdot 1^2:\mathbb{R}):\mathbb{C})\cdot\exp\bigl(((\pi(1-t):\mathbb{R}):\mathbb{C})\cdot i\bigr)=\exp(i\pi(1-t)).$$
Downstream lemmas such as zArc_eq_exp, continuous_zArc, and normSq_zArc exploit that identity.
why it matters
Anchor for the entire physical-point C11 development. Downstream uses include: continuationEdgesC_physical (continuation tuple equals the two-value hinge edges at $z=\mathrm{zArc},t$); branchRegular_fourOne_hinge (S3 inhabitation: fourOne hinge branch-regular on all of $(0,1)$, using $\mathrm{im},z>0$); denom_ne ($6z-2\neq 0$ on the whole arc because $|z|=1$ excludes $z=1/3$); continuous_zArc and continuousOn_hingeCosPath (split cosine continuous on $[0,1]$ via the cut-free Möbius form); endpoint theorems continuationEdgesC_zero / _one.
It pins the PATH-SELECTED continuation with proved branch certificate on the open arc interior (panel C11 mandate). Does not close the ledger gap wick_action_continuation_4d; that remains open pending genuine interior-hinge complex existence (C12).
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